family of polynomials with square discriminant The title pretty much sums it up: do people know of nice parametrized families of polynomials (with integer coefficients) with square discriminant. I should say that one such family consists of trinomials of the form $x^n + a x^k + b,$ where $(n, k)$ is even (there are some conditions on $a, b$ which I don't remember), but that is not satisfying, since the Galois group of such things is very special (since they have the form $f(x^2).$)
 A: You want families of polynomials with Galois group contained in $A_n$. A generic polynomial for a group $G$ is a polynomial with indeterminates with Galois group $G$ so that every extension of the base field with that Galois group arises from evaluating the indeterminates. Of course, some evaluations will have smaller Galois groups. Two examples:
$A_3: x^3-tx^2+(t-3)x+1,$ discriminant $(t^2-3t+9)^2$
$D_5: x^5+(t-3)x^4+(s-t+3)x^3+(t^2-t-2s-1)x^2+s x + t,$ discriminant $t^2(4t^5-4t^4-24st^3-...+14st-4t)^2$
A: There is a beautiful construction by Mestre [1] (see also Prop. I.5.12 in [2]) which, for fixed odd degree $n$, yields an $n+1$-parametric family of such polynomials: Let $z,t_1,t_2,\dots,t_n$ be indeterminates and $K=\mathbb Q(t_1,\dots,t_n)$ and $g(X)=(X-t_1)\dots(X-t_n)$. Then there is a polynomial $h(X)\in K[X]$ of degree $n-1$ and relatively prime to $g(X)$ such that $g(X)-zh(X)$ has Galois group $A_n$ over $K(z)$. Here $h(X)$ can be computed by solving a system of linear equations.
So specializing the $t_i$'s and $z$ in $\mathbb Q$ give polynomials over $\mathbb Q$ with square discriminants, and usually with Galois group $A_n$.
In order to handle even degree $n-1$, take $g(X)$ and $h(X)$ as above and set $F(X)=\frac{g(X)h(z)-h(z)g(X)}{X-z}$. Then $F(X)$ has Galois group $A_{n-1}$ over $\mathbb K(z)$.
[1] Mestre, Jean-François:
Constructions polynomiales et théorie de Galois. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 318–323, Birkhäuser, Basel, 1995.
[2] Malle, Gunter; Matzat, B. Heinrich:
Inverse Galois theory. 
Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1999. 
